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asp.net barcode reader free Vector Basics in Visual Studio .NET
68 Vector Basics USS Code 39 Reader In .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Make Code 39 In Visual Studio .NET Using Barcode generator for Visual Studio .NET Control to generate, create Code39 image in .NET framework applications. That s the same as a b Let s call this Cartesian difference vector d = (xd,yd) Then xd = 0 and yd = 2 21/2 Using the Cartesiantopolar conversion table, we can see that qd = p /2 and rd = (xd2 + yd2)1/2 = [02 + (2 21/2)2]1/2 = 2 21/2 The polar ordered pair is therefore a b = [(p /2),(2 21/2)] The first coordinate is the angle in radians The second coordinate is the magnitude in linear units Code 39 Extended Reader In .NET Framework Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET framework applications. Paint Barcode In .NET Framework Using Barcode generation for VS .NET Control to generate, create barcode image in Visual Studio .NET applications. Are you confused
Barcode Decoder In .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Paint Code 3/9 In C# Using Barcode creation for .NET Control to generate, create Code 3/9 image in VS .NET applications. By now you might wonder, What s the difference between a polar vector sum and a Cartesian vector sum Or a polar vector negative and a Cartesian vector negative Or a polar vector difference and a Cartesian vector difference If we start with the same vector or vectors, shouldn t we get the same vector when we re finished calculating, whether we do it the polar way or the Cartesian way That s an excellent question The answer is yes The mathematical methods differ, but the resultant vectors are equivalent whether we work them out the polar way or the Cartesian way Code39 Generator In Visual Studio .NET Using Barcode maker for ASP.NET Control to generate, create Code 39 Full ASCII image in ASP.NET applications. Creating Code 3 Of 9 In Visual Basic .NET Using Barcode creation for .NET framework Control to generate, create Code39 image in .NET framework applications. Here s a challenge! Creating Code 128 Code Set C In VS .NET Using Barcode creation for .NET framework Control to generate, create Code 128B image in .NET applications. Make Bar Code In VS .NET Using Barcode generation for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. Draw polar coordinate diagrams of the vector addition and subtraction facts we worked out in this section Draw GS1 DataBar Limited In .NET Framework Using Barcode maker for .NET framework Control to generate, create GS1 DataBar Stacked image in VS .NET applications. MSI Plessey Generator In .NET Using Barcode creation for Visual Studio .NET Control to generate, create MSI Plessey image in .NET applications. Solution
UPC  13 Maker In .NET Using Barcode creator for ASP.NET Control to generate, create EAN13 image in ASP.NET applications. UPC Code Creator In C# Using Barcode drawer for .NET framework Control to generate, create UPC Symbol image in .NET applications. The original two polar vectors were a = (qa,ra) = (p /4,2) and b = (qb,rb) = (7p /4,2) Making EAN 128 In VB.NET Using Barcode generation for .NET Control to generate, create GS1 128 image in .NET applications. EAN / UCC  13 Decoder In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. The Polar Way
ANSI/AIM Code 39 Creator In C# Using Barcode encoder for VS .NET Control to generate, create USS Code 39 image in .NET applications. Generate GS1128 In VS .NET Using Barcode generator for ASP.NET Control to generate, create EAN / UCC  13 image in ASP.NET applications. We found their polar sum to be a + b = [0,(2 21/2)] and their polar difference to be a b = [(p /2),(2 21/2)] When we converted the two vectors to Cartesian form, we got a = (21/2,21/2) and b = (21/2, 21/2) We found their Cartesian sum to be a + b = [(2 21/2),0] and their Cartesian difference to be a b = [0,(2 21/2)] We can illustrate the original vectors, the vector sum, the negative of the second vector, and the vector difference in four diagrams: EAN 13 Generator In None Using Barcode printer for Software Control to generate, create EAN / UCC  13 image in Software applications. EAN 13 Drawer In Java Using Barcode printer for Java Control to generate, create EAN 13 image in Java applications. Figure 47 shows the polar sum, including a, b, and a + b
Figure 47 Polar sum of two vectors Each radial
division represents 1/2 unit
70 Vector Basics
Figure 48 shows the polar difference, including a, b, b, and a b
p /2 a b = a + ( b) = [p /2, (2 21/2)] b a = (p /4, 2) b = (7p /4, 2) This vector is converted to Cartesian form in the subtraction process
3p /2 Figure 48 Polar difference between two vectors Each radial
division represents 1/2 unit
Figure 49 shows the Cartesian sum, including a, b, and a + b
a = (21/2, 21/2) a + b = [(2 21/2), 0] x
Each axis division is 1/2 unit
b = (21/2, 21/2) Figure 49 Cartesian sum of two vectors Each axis
division represents 1/2 unit
Practice Exercises
Figure 410 shows the Cartesian difference, including a, b, b, and a b
a b = a + ( b) = [0, (2 21/2)] b = ( 21/2, 21/2) a = (21/2, 21/2) Each axis division is 1/2 unit
b = (21/2, 21/2) Figure 410 Cartesian difference between two vectors Each axis division represents 1/2 unit
Practice Exercises
This is an openbook quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find workedout answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Consider two vectors a and b in the Cartesian plane, with coordinates defined as follows: a = ( 3,6) and b = (2,5) Work out, in strict detail, the Cartesian vector sums a + b, b + a, a b, and b a 2 A vector is defined as the zero vector (denoted by a bold numeral 0) if and only if its magnitude is equal to 0 In the Cartesian plane, the zero vector is expressed as the ordered pair (0,0) Show that when a vector is added to its Cartesian negative in either order, the result is the zero vector 72 Vector Basics
3 Imagine two arbitrary vectors a and b in the Cartesian plane, with coordinates defined as follows: a = (xa,ya) and b = (xb,yb) Show that the vector b a is the Cartesian negative of the vector a b 4 Find the Cartesian sum of the vectors a = (4,5) and b = ( 2, 3) Compare this with the sum of their negatives a = ( 4, 5) and b = (2,3) 5 Prove that Cartesian vector negation distributes through Cartesian vector addition That is, show that for two Cartesian vectors a and b, it s always true that (a + b) = a + ( b) 6 Find the polar sum of the vectors a = (p /2,4) and b = (p,3) 7 Find the polar negative of the vector a + b from the solution to Problem 6 8 Find the polar negatives a and b of the vectors stated in Problem 6 9 Find the polar sum of the vectors a and b from the solution to Problem 8 Compare this with the solution to Problem 7 10 Find the polar differences a b and b a between the vectors stated in Problem 6

